A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation

In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions for nonlinear variable-order time fractional two-dimensional (2D) Schrödinger equation. First, the variable-order time fractional derivative involved in the considered problem is approximated via the finite difference technique. Then, by help of the finite difference scheme and the theta-weighted method, a recursive algorithm is derived for the problem under examination. After that, the real functions available in the real and imaginary parts of the unknown solution of the problem are expanded via the 2D LWs. Finally, by applying the operational matrices of derivative, the solution of the problem is transformed to the solution of a linear system of algebraic equations in each time step which can simply be solved. In the proposed method, acceptable approximate solutions are achieved by employing only a small number of the basis functions. To illustrate the applicability, validity and accuracy of the wavelet method, some numerical test examples are solved using the suggested method. The achieved numerical results reveal that the method established based on the 2D LWs is very easy to implement, appropriate and accurate in solving the proposed model.